How do I add polygons to fill a face?
Select all the vertices on the edge and do Grid fill



Then select a few vertices on the center, enable proportional editing and pull them to create the shape you need.

In your case there is a big ugly n-gon that you need to delete in order to make grid fill work.



Now the grid fill operation also makes another issue evident in your model: the subdivisions and number of vertices are not the same on both sides of your model. Undo the grid fill, add loop cuts to fix the assymmetry making both side of the model have the same number of loop cuts. Now  select the edges and do the grid fill.  Your model should work just fine now.



Then just enable proportional editing to create the bulge you are after.